Why does Vr increase with lower air density and decrease with higher air density?

Question

Can someone explain why Vr increase with lower air density and decrease with higher air density? I read such a phrase in my ATPL(A) performance textbook, but without further explanation. As far as I know Vr is an indicated airspeed, so it should be independent from air density.

Answer 1

Part of Part 25's definition of VR is:

The speed (determined in accordance with § 25.111(c)(2)) that allows reaching V2 before reaching a height of 35 feet above the takeoff surface

At lower air density the engine produces less thrust (less air to burn), so to arrive at the same height at the same location while reaching V2, a better lift to drag ratio is needed, i.e. same lift with less drag, which needs a slightly faster speed to allow a smaller angle of attack.

To put that into perspective, for the Boeing 737-800 the increment is approximately one knot for every 2,000 feet increase in elevation:

Lettered areas are the corrections A through E:

And above are example numbers for the correction columns A through E. Very subtle.